Triple integrals in cylindrical and spherical coordinates
Triple integrals in cylindrical and spherical coordinates
Triple integrals in cylindrical and spherical coordinates
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MATH 209—<br />
Calculus,<br />
III<br />
Volker Runde<br />
<strong>Triple</strong><br />
<strong><strong>in</strong>tegrals</strong> <strong>in</strong><br />
cyl<strong>in</strong>drical<br />
coord<strong>in</strong>ates<br />
<strong>Triple</strong><br />
<strong><strong>in</strong>tegrals</strong> <strong>in</strong><br />
<strong>spherical</strong><br />
coord<strong>in</strong>ates<br />
Integration <strong>in</strong> <strong>spherical</strong> coord<strong>in</strong>ates, II<br />
As <strong>in</strong> rectangular coord<strong>in</strong>ates. . .<br />
Divide E <strong>in</strong>to small <strong>spherical</strong> wedges Ej,k,ℓ, <strong>and</strong> pick a support<br />
po<strong>in</strong>t (x ∗ j,k,ℓ , y ∗<br />
j,k,ℓ , z∗ j,k,ℓ ) ∈ Ej,k,ℓ.<br />
Then:<br />
���<br />
f (x, y, z) dV<br />
with<br />
E<br />
= lim<br />
n,m,ν→∞<br />
n�<br />
m�<br />
ν�<br />
j=1 k=1 ℓ=1<br />
∆Vj,k,ℓ = volume of Ej,k,ℓ.<br />
f (x ∗ ∗<br />
j,k,ℓ , yj,k,ℓ , z∗ j,k,ℓ )∆Vj,k,ℓ
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